// Numbas version: exam_results_page_options
{"name": "Partial fraction breakdown A,B & C: repeated linear factor", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"b1": {"templateType": "randrange", "definition": "random(6..10#1)", "name": "b1", "group": "Ungrouped variables", "description": ""}, "a1": {"templateType": "randrange", "definition": "random(1..5#1)", "name": "a1", "group": "Ungrouped variables", "description": ""}, "F": {"templateType": "randrange", "definition": "random(1..6#1)", "name": "F", "group": "Ungrouped variables", "description": ""}, "H": {"templateType": "randrange", "definition": "random(2..12#1)", "name": "H", "group": "Ungrouped variables", "description": ""}}, "variable_groups": [], "functions": {}, "rulesets": {}, "preamble": {"css": "", "js": ""}, "advice": "<p>\\(Q(s)=\\frac{\\var{F}s^2+\\var{H}}{(S+\\var{a1})(s+\\var{b1})^2}=\\frac{A}{s+\\var{a1}}+\\frac{B}{s+\\var{b1}}+\\frac{C}{(s+\\var{b1})^2}\\)</p>\n<p>Multiply across by \\((s+\\var{a1})(s+\\var{b1})^2\\)</p>\n<p>\\(\\var{F}s^2+\\var{H}=A(s+\\var{b1})^2+B(s+\\var{a1})(s+\\var{b1})+C(s+\\var{a1})\\)</p>\n<p>let \\(s=-\\var{a1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{F}(-\\var{a1})^2+\\var{H}=A(\\simplify{{b1}-{a1}})^2+B(0)+C(0)\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{F}*{a1}^2+{H}}=\\simplify{({b1}-{a1})^2}A\\)</p>\n<p style=\"padding-left: 30px;\">\\(A=\\frac{\\simplify{{F}*{a1}^2+{H}}}{\\simplify{({b1}-{a1})^2}}\\)</p>\n<p>let \\(s=-\\var{b1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{F}(-\\var{b1})^2+\\var{H}=A(0)+B(0)+C(\\simplify{-{b1}+{a1}})\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\simplify{{F}*{b1}^2+{H}}=\\simplify{(-{b1}+{a1})}C\\)</p>\n<p style=\"padding-left: 30px;\">\\(C=\\frac{\\simplify{{F}*{b1}^2+{H}}}{\\simplify{(-{b1}+{a1})}}\\)</p>\n<p>Compare coefficients of \\(s^2\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{F}=A+B\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\var{F}=\\frac{\\simplify{{F}*{a1}^2+{H}}}{\\simplify{({b1}-{a1})^2}}+B\\)</p>\n<p style=\"padding-left: 30px;\">\\(B=\\simplify{(({F}*({b1}^2-2*{b1}*{a1})-{H})/({b1}-{a1})^2)}\\)</p>", "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "tags": [], "statement": "<p>Find the partial fraction breakdown of the compound fraction:</p>\n<p style=\"padding-left: 30px;\">\\(Q(s)=\\frac{\\var{F}s^2+\\var{H}}{(s+\\var{a1})(s+\\var{b1})^2}\\)</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "parts": [{"gaps": [{"mustBeReduced": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "maxValue": "({F}*{a1}^2+{H})/(({b1}-{a1})^2)", "allowFractions": true, "scripts": {}, "variableReplacements": [], "minValue": "({F}*{a1}^2+{H})/(({b1}-{a1})^2)", "correctAnswerFraction": true, "showCorrectAnswer": true, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "mustBeReducedPC": 0, "showFeedbackIcon": true}, {"mustBeReduced": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "maxValue": "{F}-({F}*{a1}^2+{H})/(({b1}-{a1})^2)", "allowFractions": true, "scripts": {}, "variableReplacements": [], "minValue": "{F}-({F}*{a1}^2+{H})/(({b1}-{a1})^2)", "correctAnswerFraction": true, "showCorrectAnswer": true, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "mustBeReducedPC": 0, "showFeedbackIcon": true}, {"mustBeReduced": false, "correctAnswerStyle": "plain", "variableReplacementStrategy": "originalfirst", "maxValue": "({F}*{b1}^2+{H})/({a1}-{b1})", "allowFractions": true, "scripts": {}, "variableReplacements": [], "minValue": "({F}*{b1}^2+{H})/({a1}-{b1})", "correctAnswerFraction": true, "showCorrectAnswer": true, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "mustBeReducedPC": 0, "showFeedbackIcon": true}], "scripts": {}, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "marks": 0, "prompt": "<p>If the partial fraction breakdown is given by:</p>\n<p style=\"padding-left: 30px;\">\\(Q(s) =\\frac{A}{s+\\var{a1}}+\\frac{B}{s+\\var{b1}}+\\frac{C}{(s+\\var{b1})^2}\\)</p>\n<p>Calculate the values of \\(A, B\\) and \\(C\\) and give your answers as fractions</p>\n<p style=\"padding-left: 30px;\">\\(A=\\) [[0]]</p>\n<p style=\"padding-left: 30px;\">\\(B=\\)&nbsp;[[1]]</p>\n<p style=\"padding-left: 30px;\">\\(C=\\)&nbsp;[[2]]</p>", "showFeedbackIcon": true, "type": "gapfill", "variableReplacements": []}], "name": "Partial fraction breakdown A,B & C: repeated linear factor", "ungrouped_variables": ["a1", "b1", "F", "H"], "extensions": [], "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}